On the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions

TL;DR

This paper proves the local-in-time well-posedness of classical solutions for the vacuum free boundary problem in the viscous Saint-Venant system for shallow water in two dimensions, addressing degeneracies at the vacuum boundary.

Contribution

It introduces a novel approach using a degenerate-singular elliptic operator and weighted energy functionals to establish higher-order regularities up to the vacuum boundary.

Findings

Established local well-posedness of classical solutions

Achieved higher-order regularities uniformly up to the vacuum boundary

Developed a new method to handle degeneracies in the equations

Abstract

In this paper, we establish the local-in-time well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow water in two dimensions. The solutions are shown to possess higher-order regularities uniformly up to the vacuum free boundary, although the depth degenerates as a singularity of the distance to the vacuum boundary. Since the momentum equations degenerate in both the dissipation and time evolution, there are difficulties in constructing approximate solutions by the Galerkin's scheme and gaining higher-order regularities uniformly up to the vacuum boundary for the weak solution. To construct the approximate solutions, we introduce some degenerate-singular elliptic operator, whose eigenfunctions form an orthogonal basis of the projection space. Then the high-order regularities on the weak solution are obtained by using some carefully designed higher-order weighted energy functional.